Internal field distribution measurement in 1-D ... - OSA Publishing

Internal field distribution measurement in 1-D strongly anisotropic sub-wavelength periodic structures of finite length Mesfin Woldeyohannes,1 John O. Schenk,2 Robert P. Ingel,2 Shawn P. Rigdon,3 Mitchell Pate,3 John D. Graham,4 Michael Clare,4 Weiguo Yang,3,∗ and Michael A. Fiddy2 1 Chemistry

and Physics Department, Western Carolina University, Cullowhee, North Carolina 28723, USA 2 Center for Optoelectronics and Optical Communications, University of North Carolina at Charlotte, Charlotte, North Carolina 28223, USA 3 Department of Engineering and Technology, Western Carolina University, Cullowhee, North Carolina 28723, USA 4 Center for Rapid Prototype and Product Realization, Kimmel School, Western Carolina University, Cullowhee, North Carolina 28723, USA ∗ [email protected]

Abstract: We report measurements of the internal field intensity distribution in finite length one dimensional strongly anisotropic sub-wavelength periodic structures in the vicinity of the photonic band gap (PBG) edge. The strong in-plane anisotropy of more than 10% index contrast is obtained via form birefringent sub-wavelength gratings. The structures have a period of less than half the wavelength. Depending on the excitation frequency, both standing wave and evanescent Bloch modes can be identified and observed experimentally. The field enhancement near the PBG edge is confirmed also but at a significantly reduced strength attributed to the small but finite material loss. © 2011 Optical Society of America OCIS codes: (160.3918) Metamaterials; (050.2555) Form birefringence; (160.5298) Photonic crystals.

References and links 1. J. D. Joannopoulos, P. R. Villeneuve, and S. Fan, “Photonic crystals,” Solid State Commun. 102(2–3), 165–173 (1997), http://www.sciencedirect.com/science/article/B6TVW-3SP68WM-7H/2/36c7658507bdbbc4a692dd9c0445f406. 2. J. D. Joannopoulos, S. G. Johnson, J. N. Winn, and R. D. Meade, Photonic crystals: molding the flow of light (second edition), 2nd ed. (Princeton University Press, 2008). 3. Y. A. Vlasov, M. O’Boyle, H. F. Hamann, and S. J. McNab, “Active control of slow light on a chip with photonic crystal waveguides,” Nature 438, 65–69 (2005). 4. Y. Akahane, T. Asano, B.-S. Song, and S. Noda, “High-Q photonic nanocavity in a two-dimensional photonic crystal,” Nature 425, 944–947 (2003). 5. H. Kosaka, T. Kawashima, A. Tomita, M. Notomi, T. Tamamura, T. Sato, and S. Kawakami, “Superprism phenomena in photonic crystals: toward microscale lightwave circuits,” J. Lightwave Technol. 17(11), 2032 (1999), http://jlt.osa.org/abstract.cfm?URI=JLT-17-11-2032. 6. M. Notomi, “Theory of light propagation in strongly modulated photonic crystals: refractionlike behavior in the vicinity of the photonic band gap,” Phys. Rev. B 62(16), 10696–10705 (2000).

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Received 13 Sep 2010; accepted 14 Oct 2010; published 21 Dec 2010

3 January 2011 / Vol. 19, No. 1 / OPTICS EXPRESS 81

7. E. Cubukcu, K. Aydin, and C. M. Ozbay, E. Foteinopoulou, and S. Soukoulis, “Electromagnetic waves: negative refraction by photonic crystals,” Nature 423(6940), 604–605 (2003). 8. A. Berrier, M. Mulot, M. Swillo, M. Qiu, L. Thyl´en, A. Talneau, and S. Anand, “Negative refraction at infrared wavelengths in a two-dimensional photonic crystal,” Phys. Rev. Lett. 93(7), 073902 (2004). 9. P. V. Parimi, W. T. Lu, P. Vodo, J. Sokoloff, J. S. Derov, and S. Sridhar, “Negative refraction and left-handed electromagnetism in microwave photonic crystals,” Phys. Rev. Lett. 92(12), 127401 (2004). 10. N. Mattiucci, G. D’Aguanno, M. Scalora, M. J. Bloemer, and C. Sibilia, “Transmission function properties for multi-layered structures: application to super-resolution,” Opt. Express 17(20), 17517–17529 (2009), http://www.opticsexpress.org/abstract.cfm?URI=oe-17-20-17517. 11. J. Ballato, A. Ballato, A. Figotin, and I. Vitebskiy, “Frozen light in periodic stacks of anisotropic layers,” Phys. Rev. E 71(3), 036612 (2005). 12. A. Figotin and I. Vitebskiy, “Frozen light in photonic crystals with degenerate band edge,” Phys. Rev. E 74(6), 066613 (2006). 13. A. A. Chabanov, “Strongly resonant transmission of electromagnetic radiation in periodic anisotropic layered media,” Phys. Rev. A 77(3), 033811 (2008). 14. Y. Cao, J. Schenk, R. P. Ingel, M. A. Fiddy, K. Burbank, M. Graham, P. Sanger, and W. Yang, “Form birefringent anisotropic photonic crystal exhibiting external field anomalies,” in Photonic Crystal Materials and Devices VII (2008). 15. J. O. Schenk, R. P. Ingel, M. A. Fiddy, and W. Yang, “Split band edge structures and negative index,” in Slow and Fast Light, p. SMB6 (Optical Society of America, 2008). 16. K. Sinchuk, R. Dudley, J. D. Graham, M. Clare, M. Woldeyohannes, J. O. Schenk, R. P. Ingel, W. Yang, and M. A. Fiddy, “Tunable negative group index in metamaterial structures with large formbirefringence,” Opt. Express 18(2), 463–472 (2010), http://www.opticsexpress.org/abstract.cfm?URI=oe-18-2-463. 17. D. Dahan and G. Eisenstein, “Tunable all optical delay via slow and fast light propagation in a Raman assisted fiber optical parametric amplifier: a route to all optical buffering,” Opt. Express 13(16), 6234–6249 (2005), http://www.opticsexpress.org/abstract.cfm?URI=oe-13-16-6234. 18. J. Sharping, Y. Okawachi, and A. Gaeta, “Wide bandwidth slow light using a Raman fiber amplifier,” Opt. Express 13(16), 6092–6098 (2005), http://www.opticsexpress.org/abstract.cfm?URI=oe-13-16-6092. 19. D. H. Raguin and G. M. Morris, “Antireflection structured surfaces for the infrared spectral region,” Appl. Opt. 32(7), 1154–1167 (1993), http://ao.osa.org/abstract.cfm?URI=ao-32-7-1154. 20. E. B. Grann, M. G. Moharam, and D. A. Pommet, “Artificial uniaxial and biaxial dielectrics with use of two-dimensional subwavelength binary gratings,” J. Opt. Soc. Am. A 11(10), 2695–2703 (1994), http://josaa.osa.org/abstract.cfm?URI=josaa-11-10-2695. 21. A. Mandatori, C. Sibilia, M. Bertolotti, S. Zhukovsky, J. W. Haus, and M. Scalora, “Anomalous phase in onedimensional, multilayer, periodic structures with birefringent materials,” Phys. Rev. B 70(16), 165107 (2004).

1.

Introduction

A one-dimensional (1-D) photonic crystal (PC) is a periodic array of alternating dielectric layers of different refractive indices exhibiting a frequency gap (a stop band). At this band gap, the propagation of electromagnetic waves in the direction of periodicity is forbidden for all polarizations [1, 2], which has found numerous and diverse potential applications in integrated optics [2–5]. Typically, the periodicity of a PC is on the order of the wavelength. Owing to their exceptional control over the dispersion relation of the light, PCs with sub-wavelength periodicity can be considered as one type of metamaterial. Both extremely slow light and effective negative index of refractions have been observed experimentally [6–10]. Fundamentally, the electromagnetic eigen-modes of such a periodic structure are Bloch waves of the form Φk (z) = eikz uk (z)

(1)

where z is the direction of propagation (perpendicular to the surface of the dielectric layers), eikz is the plane wave envelope function and uk (z) is a periodic function with period L - the period of the 1-D structure. k is the wave number, and for propagating Bloch modes, k is real and can be reduced to the first Brillouin zone k ∈ [−π /L, π /L]. Bloch modes play a critical role in all periodic systems and can be considered as the free particles of the periodic system. For example, in semiconductor crystals such as Si, the carriers, namely the ‘electrons and ‘holes, are in fact the energy quanta of excited Bloch modes. By analogy with their electronic counter

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part, the Bloch modes in photonic periodic structures such as PCs can be considered as free propagating plane waves. They possess the same properties as the plane waves in free space and uniform media including reflection, refraction, as well as interference. In a photonic periodic structure, the Bloch modes arise theoretically from the fact that the translational operator and the system Hamiltonian commute for a periodic structure. Physically, the periodicity of the structure causes coherent interference of light scattered by the layers interfaces, resulting, for as few as several unit cells, in sharp transmission peaks below the photonic band-edge frequency or energy. This phenomenon is known as a transmission band-edge resonance, and there is no transmission at all in the photonic band gap (PBG). The sharp peaks of the transmission band edge resonances can also been understood, by analogy with Fabry-Perot (FP) resonances of plane waves, as the Fabry-Perot resonances of the propagating Bloch modes. Similarly, for frequencies inside the PBG, the Bloch modes are evanescent, decaying exponentially in the z direction and characterized by an imaginary wave number k [11–13]. As for the propagating Bloch modes, these evanescent Bloch modes are analogous to the evanescent waves in free space or uniform media, like those excited in a lower index medium following total internal reflection (TIR). For a finite length PC structure, when propagating waves at the forbidden frequencies are impinging upon the structure, they will be totally reflected in a way similar to TIR and excite the evanescent Bloch modes inside the structure. The evanescent Bloch modes do not carry energy flow but do possess fields and electromagnetic field energy of finite strength within a short distance on the order of one period inside the structure. These analogies of propagating Bloch modes to plane waves and evanescent Bloch modes to evanescent waves resulting from TIR are useful. They are helpful in understanding the nature of both propagating and evanescent waves as well as for potential applications including super resolution imaging using PC or metamaterial structures. A direct measurement of the electromagnetic field inside PC structures over scales of the order of the wavelength is extremely difficult in optical frequencies. No direct experimental observations of the internal field distribution have been reported to the best of our knowledge for any PC or metamaterial structures. In this paper, we report direct measurements of the internal field intensity distribution over a finite length one dimensional strongly anisotropic sub-wavelength periodic structure in the vicinity of the photonic band gap edge, probing for the first time experimentally and directly the propagating Bloch modes as well as the evanescent Bloch modes. The strong anisotropy of more than 10% in TE/TM index contrast is obtained by form birefringence using sub-wavelength gratings. We have implemented these anisotropic sub-wavelength periodic structures in X-band (8-12 GHz) not only to establish the feasibility of direct field measurements on the order of wavelength but also for potential wireless and satellite communication applications [14, 15]. 2.

Design and simulation

Consider a periodic stack of anisotropic layers with a finite number of periods and having misaligned in-plane anisotropy in each period. The internal field enhancement at transmission band-edge resonance frequencies near the degenerate band-edge (DBE) is predicted [11, 12] to be significantly stronger than that achievable with more typical periodic stacks of isotropic layers. Using isotropic layers the corresponding and so called regular band-edge (RBE) leads to field enhancements which have a second order dependence on the number N of periods, whereas the DBE transmission resonances exhibit an internal field enhancement proportional to the fourth power of N. Experimentally, exceptionally high Q transmission resonances have been observed with N as small as 12 near a split band edge (SBE) where the photonic band edge split into two parts [11, 13–15]. Theoretically, SBE transmission resonances exhibit an internal field enhancement that could still go as the fourth power of N. Strong DBE or SBE

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transmission resonances promise drastically reduced footprints for many applications exploiting transmission band-edge resonance effects including slow light optical tunable delay lines, enhanced optical nonlinearity, and improved sensitivity for wireless and satellite communications [10, 14–18]. We have implemented here a strong anisotropic sub-wavelength periodic structure with a designed common PBG near 10 GHz for both transverse electrical (TE) and transverse magnetic (TM) polarizations. The large difference between the indices of refraction for TE and TM polarizations is achieved by form birefringence [16, 19, 20]. Form birefringence provides the index anisotropy through sub-wavelength patterning of a host medium. For sub-wavelength gratings, the effective dielectric constants are given by [19, 20] 2 π2 Λ 2 2 0 2 (εs − εo ) εT E = ε T E 1 + f (1 − f ) (2) 3 λ εo εT0 E 0 ε0 π2 Λ 2 2 0 2 (εs − εo ) εT E TM (3) εT M = ε T M 1 + f (1 − f ) 3 λ εo εo εs where and

εT0 E = f εs + (1 − f )εo 1

εT0 M

=

f 1− f + . εs εo

(4) (5)

In the above formulas, εs is the dielectric constant of the host material. εo is the dielectric constant of the filling material, which is usually just air so εo = 1. f is the fill factor that equals to the volume ratio between host material and filling material. λ is the wavelength of the light and Λ < λ is the sub-wavelength grating period. The TE polarization is parallel to the grating lines, and the TM polarization is perpendicular to the grating lines. In our design, we have used the engineering polymer acrylonitrile butadiene styrene, known as the ABS plastic, as the host material. The dielectric constant of the ABS plastic in X-band is measured as εs = 2.47 using the X-band waveguide architecture together with a network analyzer (Agilent Technology N5230A) and the standard material characterization software (Agilent Technology 85071E). The loss tangent is measured to be less than 5e-4. Figure 1 shows the predicated effective dielectric constants v.s. the fill factor. The markers in the figure shows the measured effective dielectric constants for sub-wavelength grating samples with f = 0.5 and period of 1 mm. The agreement between the prediction and the experimental results is satisfactory and the measured TE and TM indices are nT E = 1.30 and nT M = 1.17 giving an index contrast of Δn/n =10.5%. Such a large birefringence is rare in naturally occurring materials. For example, calcite, which is considered one of the strongest anisotropic materials, has an index contrast of about 10%. The index contrast of the sub-wavelength gratings can be enhanced by several means including surface treatments and material doping, giving index contrasts as high as 70% [16]. Figure 2(a) shows a sub-wavelength grating disc and Fig. 2(b) shows the building block used for the periodic stack of anisotropic layers. Using the finite element method (FEM), we have analyzed the spatial modes of the structure assuming aligned birefringent layers. Although the structure will support multiple modes, only the symmetric modes will be efficiently excited. As is evident in Fig. 3, only the nearly linear polarized modes (LP modes) where the electrical fields are predominantly either parallel or perpendicular to the sub-wavelength grating lines are symmetrical and will maximally overlap and couple with the incident beam, therefore approximating a single component plane wave propagation in infinitely large structures.

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Effective Dieletric Constants

2.5

2

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Fig. 1. Effective dielectric constants of form birefringence from sub-wavelength gratings. Solid line: TE polarization that is parallel to the grating lines; Dashed line: TM polarization that is perpendicular to the grating lines.

(a)

(b)

Fig. 2. Samples made of sub-wavelength grating structures utilizing form-birefringence. (a) 4 inch disc with 500 μ m period. (b) Two layers of sub-wavelength gratings (500 μ m period) misaligned with 45 degrees.

Under this preferred symmetric excitation condition, we can use standard transfer matrix based algorithms to simulate the anisotropic layered structure with arbitrary misalignment angles between the anisotropic layers in each period [11, 14]. The simulated transmission spectra for the structure of 51 periods with a misalignment angle of 45 degree are shown in Fig. 4. All combinations of launching and receiving polarization are considered. For example in the figure, the curve labeled as TE-TM denotes the TE polarization for launching and TM polarization for receiving. As shown in the figure, a common bandgap for both TE and TM polarizations can be seen. The unit cell length is 12.2 mm and the air gap ratio of 0.167. This results in a common bandgap for both polarizations between 10.2 GHz and 10.8 GHz. The lower band edges for TE and TM transmission spectra are 9.7 GHz and 10.2 GHz respectively. For structures with a finite number of periods, the actual band-edge resonant frequency differs from the theoretical prediction for longer structures and the deviation for structures with a smaller number of periods can be significant. This is shown in Fig. 5. Since the band edge resonance is very sharp, i.e., a narrow bandwidth effect, the structures length dependent resonant frequency is critical to the experimental investigation as well as the application of the band edge behavior.

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(a)

(b)

(c)

(d)

Fig. 3. Multiple modes supported by the anisotropic structure with zero misalignment angle between anisotropic layers within each period. Only the LP modes are symmetric and will be excited efficiently. The arrow shows the direction of electrical field oscillation.

3.

Rapid prototype fabrication

The anisotropic layered structures discussed here were fabricated using in-house rapid prototyping facilities. Rapid prototyping in a general sense refers to an automated process whereby digital representations of desired patterns and structures are created using computer aided design (CAD) software and fabricated by either an additive or a subtractive process. The subtractive process is more similar to traditional milling machine processes where one starts with a bulk material and the structures are constructed by removing material. For additive processes, the digital models are sliced into thin horizontal layers and then built up layer by layer by adding material. We used a fused deposition additive process to fabricate the sample structures. It is an extrusion process in which a thermoplastic monofilament travels through a heated tip and follows software-created toolpaths to form each layer as a build platform moves down to prepare for building the next layer. This technology uses actual engineering polymers such as ABS, polycarbonate, polyphenylsulfone (PPSF), and Ultem 9085TM . We use a Stratasys FDM Titan©rapid prototype tool. Parts up to 400 mm x 350 mm x 400 mm can be built in this machine, and parts that exceed the size of the Titans build envelope may be built in sections and

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1 TE−TM TM−TE

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assembled to form a complete unit. Models are produced within an accuracy of ±0.127 mm up to 127 mm. Accuracy on models greater than 127 mm is ±1.5 μ m per mm. Ideal wall thickness of ABS and polycarbonate is 1 mm. The minimum wall thickness with any material is approximately 0.5 mm. Parts built in the Titan are 60% to 80% as strong as those made of injection molded plastic. Figure 6(a) shows the numerical parametric model of a single period building block and Figure 6(b) shows the assembled form-birefringent periodic structure where the unit cells are fabricated using the Stratasys FDM Titan©. The probe holes are built into the CAD model and were incorporated to facilitate the internal field measurements.

(a)

(b)

Fig. 6. (a) Parametric model of the unit cell and (b) the assembled form-birefringent periodic structure.

The benefit of using parametric modeling is that the structure can be scaled up or down for operations at different wavelength ranges. While the Titan FDM tool is suitable for fabricating RF and microwave structures, other rapid prototyping tools including direct laser micromachining are available for fabricating structures requiring smaller feature sizes down to sub-microns. 4.

Experiment setup and results

The experimental setup is shown in Fig. 7. The transmitting and receiving horn antennas have a Vector Network Analyzer (VNA)

DUT Horn Antenna

Horn Antenna

Fig. 7. Experiment setup. VNA: Vector Network Analyzer. DUT: Device Under Test.

nominal directivity gain of 17 dBi. Accordingly, only modes with small divergence are excited. #133651 - $15.00 USD

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Also in the figure, the vector network analyzer (VNA, Agilent Technology N5230A) is used to generate the X-band signal and excite the anisotropic periodic structure at total of 6401 discrete dwelled frequencies from 9 GHz to 11.5 GHz, which covers a range from below the lower band edges (TE LBE around 9.6 GHz) to well into the bandgap. Figure 8 shows the measured transmission spectra, which are qualitatively in agreement with the simulation.

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(b)

Fig. 8. Measured transmission spectra. (a) Cross polarization configuration. (b) Parallel polarization configuration.

The common bandgap from 10.5 GHz to 11 GHz for both TE and TM polarizations is evident from the results. This compares favorably to the expected common bandgap from 10.2 GHz to 10.8 GHz. The discrepancy can be attributed to uncertainties in actual material parameters and also partly explained by the bandtail effect caused by the small variations in the thicknesses of the layers. The less pronounced FP fringes in the measured spectra as compared to the simulation are the result of the non-zero divergence of the beam launched and received from the horn antennas. This is similar to a loss of interference fringe contrasts due to finite bandwidth. The band edge behavior is more pronounced in the phase spectra as shown in Fig. 9 which shows the transmission phase response in the vicinity of the TE lower band edge. The clear appearance of the band-edge modified dispersion verifies good periodicity. As can be seen from the Fig. 9, the calibrated spectral phase deviates from a linear dispersion relation and exhibits the characteristic parabolic feature of the dispersion relation near a band edge around 9.67 GHz. The TM-TM spectrum also exhibits a portion of spectral phase that exhibits the anomalous behavior of a negative group index with a positive slope near 9.7 GHz, which is not unusual for anisotropic layered structures [16, 21]. In order to measure the field distribution along the finite length structure, a coaxial cabled connected mono-pole probe was placed at two orthogonal positions at each period picking up the electric field strength of both TE (vertical) and TM (horizontal) polarizations. Accordingly, the field distribution over the length of the structure is measured in the frequency range with an ultra-fine frequency step of 4x10− 5 of the center frequency. Figure 10 shows comparison of normalized field distributions based on simulations of the ideal lossless structure and actual measurements at different frequencies. Referring to the figure, the frequencies are respectively: (a) 9.58 GHz and (b) 9.65 GHz, which are outside the forbidden band; (c) 9.67 GHz, which is in the vicinity of the TE band edge; (d) 10.20, GHz which is in the vicinity of the TM band edge; (e) 10.25 GHz, which is just inside the common band gap; and (f) 10.5 GHz, which is well inside the common forbidden band. The simulated trace (solid lines) shows the total field #133651 - $15.00 USD

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100 period Transmission Phase (Cal.ed) 3

Tra ansmission n Phase

2 1 0 -1 1 TETM TMTM

-2 -3 9.4

9.5

9.6 9.7 9.8 RF Frequency (GHz)

9.9

Fig. 9. Measured transmission spectral phase.

intensity distribution averaged over a unit cell. The measured data (circled lines) are electric field intensity probed at a single position per unit cell. Taking into account the local variation of field distribution resulting from the non-plane wave part of the total Bloch wave function, the measurement results can be seen as in good agreement with the simulations. However, the simulation predicts a maximum total field enhancement of 101.8 at the TE LBE transmission resonance while the measured result only shows an electric field enhancement of 6.82 referenced to the input. This reduction in field enhancement is attributed to the material loss, albeit as small as less than 0.1 percent in loss tangent. Outside the photonic band gap, the forward and backward propagating Bloch waves interfere with each other and build up a standing Bloch wave as shown in Fig. 10(a) and 10(b). The field distribution inside the periodic structure is oscillating similar to that of FP modes inside a cavity. The closer to the band edge, the smaller the Bloch mode indices and the larger the separations between field interference or enhancement maxima. In the vicinity of the transmission band edge, where the effective Bloch mode index is equal to one, the effective wavelength of the Bloch mode is that of the fundamental mode of the FP cavity of the structures length. The single peaked distribution as shown in Fig. 10(c) indeed manifests the fundamental Bloch mode of the structure for the TE polarization. Similar behavior also occurs at the TM band edge of 10.2 GHz as shown in Fig. 10(d), where a single peaked field distribution corresponds to the fundamental Bloch mode of the structure for the TM polarization. As we go further into the photonic band gap, the Bloch waves become evanescent gradually as shown in Fig. 10(e) and 10(f), where in (e) the field distribution has just started to become evanescent and in (f) the field distribution has become clearly evanescent. It is noted here that the non-planewave part of the Bloch mode wave function also changes with the frequency, contributing to the more complex field distribution as probed at discrete positions along the structure. 5.

Discussions and conclusions

We measured the internal field intensity distribution in strongly anisotropic 1-D sub-wavelength periodic structures of finite length. The frequency dependent field distribution agrees qualitatively with the simulations. Both standing waves and evanescent Bloch modes can be identified and observed in the experiments. The internal field enhancement due to transmission band-edge resonances is verified but at a significantly reduced strength which we attribute to the small but finite material loss. However, as can be seen from the results reported here, the Bloch modes

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9.650 GHz Normalized Field Enhancement Distribution

Normalized Field Enhancement Distribution

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Fig. 10. Measured field intensity distribution along the device at different frequencies. (a) and (b) outside the forbidden band. (c) 9.67 GHz, which is in the vicinity of the lower TE band edge; (d) 10.20GHz, which is close to the lower TM band edge; (e) 10.25 GHz, which is just inside the forbidden band; and (f) 10.50 GHz, which is well inside the forbidden band. Solid lines: simulation; circled lines: measured data

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in photonic periodic structures bear striking similarities to planewaves in free space or uniform media in the exactly same way as the carriers in semiconductors do to free electrons in vacuum. The capability of manipulating these modes using sub-wavelength periodic structures or metamaterials provides possibilities for many photonic applications including super-resolution imaging. Acknowledgements MAF, JOS, and RPI acknowledge the support of the Department of Energy DE-FG0206CH11460 and Defense Advanced Research Projects Agency (DARPA) HR0011-08-C-0088. MAF also acknowledges support of National Research Foundation of Singapore (NRF) NRFGCRP 2007-01.

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